

'''simple docstring'''
class 					_snake_case		:






			def __init__(  self			,_snake_case = ""			,_snake_case = False     ):
				# Mapping from the first character of the prefix of the node
				UpperCAmelCase_     :      dict[str, RadixNode]						=  {}

				# A node will be a leaf if the tree contains its word
				UpperCAmelCase_     :      List[Any]						=  is_leaf

				UpperCAmelCase_     :      Union[str, Any]						=  prefix






			def 		UpperCamelCase__							(  self			,_snake_case     ):
				UpperCAmelCase_     :      Optional[Any]						=  0
				for q, w in zip(self.prefix			,_snake_case     ):
					if q != w:
						break

					x += 1

				return self.prefix[:x], self.prefix[x:], word[x:]






			def 		UpperCamelCase__							(  self			,_snake_case     ):
				for word in words:
					self.insert(_snake_case     )






			def 		UpperCamelCase__							(  self			,_snake_case     ):
				# Case 1: If the word is the prefix of the node
				# Solution: We set the current node as leaf
				if self.prefix == word:
					UpperCAmelCase_     :      Optional[int]						=  True

				# Case 2: The node has no edges that have a prefix to the word
				# Solution: We create an edge from the current node to a new one
				# containing the word
				elif word[0] not in self.nodes:
					UpperCAmelCase_     :      List[Any]						=  RadixNode(prefix=_snake_case			,is_leaf=_snake_case     )

				else:
					UpperCAmelCase_     :      str						=  self.nodes[word[0]]
					UpperCAmelCase_					,					UpperCAmelCase_					,					UpperCAmelCase_     :      str						=  incoming_node.match(
					    _snake_case     )

					# Case 3: The node prefix is equal to the matching
					# Solution: We insert remaining word on the next node
					if remaining_prefix == "":
						self.nodes[matching_string[0]].insert(_snake_case     )

					# Case 4: The word is greater equal to the matching
					# Solution: Create a node in between both nodes, change
					# prefixes and add the new node for the remaining word
					else:
						UpperCAmelCase_     :      str						=  remaining_prefix

						UpperCAmelCase_     :      Optional[int]						=  self.nodes[matching_string[0]]
						UpperCAmelCase_     :      int						=  RadixNode(_snake_case			,_snake_case     )
						UpperCAmelCase_     :      Dict						=  aux_node

						if remaining_word == "":
							UpperCAmelCase_     :      Union[str, Any]						=  True
						else:
							self.nodes[matching_string[0]].insert(_snake_case     )






			def 		UpperCamelCase__							(  self			,_snake_case     ):
				UpperCAmelCase_     :      Optional[Any]						=  self.nodes.get(word[0]			,_snake_case     )
				if not incoming_node:
					return False
				else:
					UpperCAmelCase_					,					UpperCAmelCase_					,					UpperCAmelCase_     :      Optional[int]						=  incoming_node.match(
					    _snake_case     )
					# If there is remaining prefix, the word can't be on the tree
					if remaining_prefix != "":
						return False
					# This applies when the word and the prefix are equal
					elif remaining_word == "":
						return incoming_node.is_leaf
					# We have word remaining so we check the next node
					else:
						return incoming_node.find(_snake_case     )






			def 		UpperCamelCase__							(  self			,_snake_case     ):
				UpperCAmelCase_     :      str						=  self.nodes.get(word[0]			,_snake_case     )
				if not incoming_node:
					return False
				else:
					UpperCAmelCase_					,					UpperCAmelCase_					,					UpperCAmelCase_     :      str						=  incoming_node.match(
					    _snake_case     )
					# If there is remaining prefix, the word can't be on the tree
					if remaining_prefix != "":
						return False
					# We have word remaining so we check the next node
					elif remaining_word != "":
						return incoming_node.delete(_snake_case     )
					else:
						# If it is not a leaf, we don't have to delete
						if not incoming_node.is_leaf:
							return False
						else:
							# We delete the nodes if no edges go from it
							if len(incoming_node.nodes     ) == 0:
								del self.nodes[word[0]]
								# We merge the current node with its only child
								if len(self.nodes     ) == 1 and not self.is_leaf:
									UpperCAmelCase_     :      Dict						=  list(self.nodes.values()     )[0]
									UpperCAmelCase_     :      str						=  merging_node.is_leaf
									self.prefix += merging_node.prefix
									UpperCAmelCase_     :      List[Any]						=  merging_node.nodes
                    # If there is more than 1 edge, we just mark it as non-leaf
							elif len(incoming_node.nodes     ) > 1:
								UpperCAmelCase_     :      Tuple						=  False
							# If there is 1 edge, we merge it with its child
							else:
								UpperCAmelCase_     :      Any						=  list(incoming_node.nodes.values()     )[0]
								UpperCAmelCase_     :      Dict						=  merging_node.is_leaf
								incoming_node.prefix += merging_node.prefix
								UpperCAmelCase_     :      Union[str, Any]						=  merging_node.nodes

							return True






			def 		UpperCamelCase__							(  self			,_snake_case = 0     ):
				if self.prefix != "":
					print("-" * height			,self.prefix			,"  (leaf)" if self.is_leaf else ""     )

				for value in self.nodes.values():
					value.print_tree(height + 1     )




def  a__       (    )     ->					bool:


	"""simple docstring"""





	UpperCAmelCase_     :      List[Any]						=  "banana bananas bandana band apple all beast".split()
	UpperCAmelCase_     :      Union[str, Any]						=  RadixNode()
	root.insert_many(_SCREAMING_SNAKE_CASE				)

	assert all(root.find(_SCREAMING_SNAKE_CASE				) for word in words				)
	assert not root.find("bandanas"				)
	assert not root.find("apps"				)
	root.delete("all"				)
	assert not root.find("all"				)
	root.delete("banana"				)
	assert not root.find("banana"				)
	assert root.find("bananas"				)

	return True


def  a__       (    )     ->					None:


	"""simple docstring"""





	assert test_trie()


def  a__       (    )     ->					None:


	"""simple docstring"""





	UpperCAmelCase_     :      Any						=  RadixNode()
	UpperCAmelCase_     :      str						=  "banana bananas bandanas bandana band apple all beast".split()
	root.insert_many(_SCREAMING_SNAKE_CASE				)

	print("Words:"	,				_SCREAMING_SNAKE_CASE				)
	print("Tree:"				)
	root.print_tree()


if __name__ == "__main__":
					main()





from math import isqrt, loga


def 				lowerCAmelCase__       (			a__ )      ->list[int]:




  '''simple docstring'''

  _UpperCamelCase           =    [True] * max_number
  for i in range(2    ,     isqrt(max_number - 1 ) + 1 ):
    if is_prime[i]:
      for j in range(i**2    ,     a__    ,     a__ ):
        _UpperCamelCase           =    False

  return [i for i in range(2    ,     a__ ) if is_prime[i]]


def 				lowerCAmelCase__       (			a__ = 800_800    ,     a__ = 800_800 )      ->int:




  '''simple docstring'''

  _UpperCamelCase           =    degree * loga(a__ )
  _UpperCamelCase           =    int(a__ )
  _UpperCamelCase           =    calculate_prime_numbers(a__ )

  _UpperCamelCase           =    0
  _UpperCamelCase           =    0
  _UpperCamelCase           =    len(a__ ) - 1
  while left < right:
    while (
        prime_numbers[right] * loga(prime_numbers[left] )
        + prime_numbers[left] * loga(prime_numbers[right] )
        > upper_bound
    ):
      right -= 1
    hybrid_integers_count += right - left
    left += 1

  return hybrid_integers_count


if __name__ == "__main__":
   print(F"{solution() = }")




